Quantum Heat Transformers

TL;DR

The paper introduces quantum heat transformers (QHTs) as autonomous quantum devices that regulate heat flow between two thermal junctions, creating a quantum analogue of classical absorption heat transformers and electrical transformers. It develops self-contained three-qubit models with interaction Hamiltonians (e.g., $\\mathcal{H}_{\text{int}}^{1}$ and $\\mathcal{H}_{\text{int}}^{2}$) under GKSL dynamics, showing that the secondary temperature gradient $\\Delta T_s$ can be engineered relative to the primary gradient $\\Delta T_p$ via intrinsic energy parameters, producing either step-down or step-up operation. A key finding is the necessarily transient step-down transformer, where step-down behavior arises in a transient window even in a steady-step-up setup, and this transient window can be tuned by initial qubit temperatures; the study extends to a four-qubit, self-contained QHT that enables dual-mode operation within the same setup by adjusting initial conditions. The results establish a framework for autonomous quantum thermal management with potential realizations in nanoscale platforms, contributing to decoherence control and thermal management in quantum technologies while highlighting rich dynamical regimes beyond classical transformer behavior.

Abstract

We propose a quantum heat transformer (QHT), a quantum thermodynamic device that modulates temperature gradients between two thermal junctions in quantum systems. Functionally, the QHT is analogous to classical absorption heat transformers in its ability to redistribute thermal energy without external work input. Moreover, we show that its performance ratio mirrors that of classical voltage transformers, where the intrinsic parameters of the system play a role similar to the coil turn ratios. We initially design the device for a three-qubit system, representing the smallest possible self-contained heat transformer model. Subsequently we extend to four-qubit systems, with a specific emphasis on exploring the step-down mode as the primary focus. We showcase the versatility and adaptability of the models by illustrating that a variety of self-contained setups can be constructed, each corresponding to different configurations of the interaction Hamiltonian and their associated self-contained conditions. An important effect in this study is the proof of existence of a necessarily transient step-down quantum heat transformer, that has a dual-mode characteristic, wherein the desired step-down mode can be realized within the transient regime of an originally designed step-up mode of the QHT. We also investigate how to control this transient domain up to which the necessarily transient mode can be achieved, by regulating the initial temperature of the qubits in the four-qubit settings. Therefore, this quantum heat transformer model not only acts as an analog to the classical transformers, but also enjoys advanced characteristics, enabling it to function in both step-up and step-down modes within the same setup, unattainable for classical transformers.

Figures (14)

• Figure 1: Schematic diagram showing the operation of an absorption heat transformer. An Absorption Heat Transformer takes waste heat (which is usually discarded) and converts part of it to a higher temperature, making it usable again—without using much electricity. It works on a thermodynamic cycle involving absorption and desorption processes. In the diagram waste heat at intermediate temperature ($T_i$) is partially upgraded to higher temperature ($T_h$) output, while rejecting the remaining heat at a lower temperature ($T_c$) , satisfying $T_h > T_i > T_c$.
• Figure 2: Schematic diagram of a three-qubit QHT. Here, $q_1$, $q_2$, and $q_3$ represent the three qubits of the quantum transformer, each immersed in a separate thermal bath at dimensionless temperatures $\tau_1$, $\tau_2$, and $\tau_3$, respectively. The pair of qubits $q_1$ and $q_2$ forms the primary thermal junction, while the pair $q_2$ and $q_3$ constitutes the secondary thermal junction. At any given time $t$, the temperatures of these qubits are represented as $T_1(t)$, $T_2(t)$, and $T_3(t)$, where $\tau_j=\frac{k_{B}}{\mathcal{K} } \Tilde{\tau}_j$ and $T_j(t)= \frac{k_{B}}{\mathcal{K} } \Tilde{T}_j(t)$. The temperature gradient across the primary thermal junction is defined as $\Delta T_p(t) = |T_1(t) - T_2(t)|$, while across the secondary junction it is denoted as $\Delta T_s(t) = |T_3(t) - T_2(t)|$.
• Figure 3: Time evolution of temperature gradients at primary and secondary thermal junctions, with the variation in the initial temperature of $q_3$ for the interaction Hamiltonian $\mathcal{H}_{\text{int}}^1$. In each panel, the temperature gradient of the primary thermal junction, $\Delta T_p(t)$, is represented by red lines, while that of the secondary thermal junction, $\Delta T_s(t)$, is denoted by black lines. The insets illustrate the initial behaviors of the temperature gradients. As the red curves lie above the black ones, it indicates that the QHTs are operating as step-down transformers. Here, we set $\mathcal{E}_1=1.0$, $\mathcal{E}_2=2.0$, $\tau_1=1.0$, and $\tau_2=2.0$. The self-contained condition of $\mathcal{H}_{\text{int}}^{1}$ yields $\mathcal{E}_3 = -\mathcal{E}_2 - \mathcal{E}_1$. Additionally, we choose the interaction strength $g=0.5$. The dimensionless coupling strength parameters of the Ohmic spectral density function are taken to be $\alpha_1 = 10^{-4}$, $\alpha_2 = 10^{-5}$, and $\alpha_3=10^{-3}$, while the cutoff frequencies of the baths are $\Omega_1=\Omega_2=\Omega_3=10^3$. All quantities plotted here are dimensionless.
• Figure 4: Absolute rates of change in temperatures of the three qubits with time. Here we plot the quantities $|\frac{\partial T{j}(t)}{\partial t} |$ for $j=1$, $2$, and $3$ vs. $t$ for the interaction Hamiltonian $\mathcal{H}_{\text{int}}^1$. Here, we set $\tau_3=3.0$. All other considerations are same as in Fig. \ref{['Fig:H_int_111-000']}. All the quantities plotted here are dimensionless.
• Figure 5: Time evolution of temperature gradients at primary and secondary thermal junctions, with the variation in the initial temperature of $q_3$ for the interaction Hamiltonian $\mathcal{H}_{\text{int}}^1$. Here, $\Delta T_p(t)$ is represented by red lines (solid and dashed), while $\Delta T_s(t)$ is denoted by black lines (solid and dashed). As the black curves lie above the red ones, it indicates that the QHTs are operating as step-up transformers. Here, we set $\mathcal{E}_1=-1.0$ and $\mathcal{E}_2=2.0$. The self-contained condition of $\mathcal{H}_{\text{int}}^{1}$ yields $\mathcal{E}_3 = -\mathcal{E}_2 - \mathcal{E}_1$. All other parameters are same as in Fig. \ref{['Fig:H_int_111-000']}. All quantities plotted here are dimensionless.